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# Minimal Knowledge and Negation as Failure - PowerPoint PPT Presentation

Minimal Knowledge and Negation as Failure. Ming Fang 7/24/2009. Outlines. Propositional MBNF Positive MKNF General MKNF Extended MBNF with First-order Quantification Description Logics of MKNF ICs. Propositional MKNF .

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### Minimal Knowledge and Negation as Failure

Ming Fang

7/24/2009

• Propositional MBNF

• Positive MKNF

• General MKNF

• Extended MBNF with First-order Quantification

• Description Logics of MKNF

• ICs

• Built from propositional symbols (atoms) using standard propositional connectives and two modal operators B and not.

B: “knowledge operator”K

not : “assumption operator”A

• Positive: if a formula or a theory (set of formulas) does not contain the negation as failure operator not.

• Define when a positive formula F is true in a structure (I,S):

• (I,S) is a model of positive theory T if:

• (i) the axioms of T are true in (I,S)

• (ii) there is no (I’,S ’) such that S’ is a proper superset of S and the axioms of T are true in (I ’,S ’)

• S is maximized, so the believed propositions are minimized.

• General MKNF: truth will be defined by a triple (I,Sb,Sn)

• (I,S) is a model of positive theory T if:

• (i) the axioms of T are true in (I,S,S)

• (ii) there is no (I’,S’) such that S ’ is a proper superset of S and the axioms of T are true in (I,S’,S)

• An example:

• It is true in (I,S’S) when:



Then a model must satisfy:

(i)

(ii)

Three cases:

• F is tautology  M=(I,S), S is the set of all interpretations.

• F is not tautology but a logical consequence of G  no model

• F is not a logical consequence of G  M=(I,Mod(G))

• Names: object constants representing all elements of |I |

• (I,S) is a model of positive theory T if:

• (i) the axioms of T are true in (I,S,S)

• (ii) there is no (I’,S’) such that S ’ is a proper superset of S and the axioms of T are true in (I,S’,S)

• An example:

• Which courses are taught?

• Which courses are taught by known individuals?

• Goal:

• represent non-first-order features of frame systems

• A set of interpretations M is a model of Σif:

• (i) the structure (M,M) satisfies Σ

• (ii) for each set of interpretations M’, if M’M, then (M’,M) does not satisfy Σ

• An ideal rational agent trying to decide which set of propositions to believe.

• Set of prior beliefs + set of rules  new beliefs

• “logical closure”

• Deduced set of beliefs coincides with the assumed believe  assumed set is justified  candidate for the agent to believe in

• Two kinds of beliefs:

• Beliefs that the agent assumed (A operator)

• New beliefs that derived (K operator)

• Example 1

• IC: Each known employee must be known to be either male or female.

Σ = <T,A> = <{},{employee(bob)}>

• Example 1

• Example 2

• IC: Each known employee has known social security number, which is known to be valid

Σ = <T,A> = <{},{employee(bob)}>